L11a311

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_22,12,9,11 X_2,9,3,10 X_20,18,21,17 X_18,5,19,6 X_4,19,5,20 X_14,7,15,8 X_16,13,17,14 X_8,15,1,16 X_6,22,7,21
Gauss code	{1, -4, 2, -7, 6, -11, 8, -10}, {4, -1, 3, -2, 9, -8, 10, -9, 5, -6, 7, -5, 11, -3}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{3}v^{3}-3u^{3}v^{2}+3u^{3}v-u^{3}-3u^{2}v^{3}+10u^{2}v^{2}-11u^{2}v+3u^{2}+3uv^{3}-11uv^{2}+10uv-3u-v^{3}+3v^{2}-3v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {14}{q^{9/2}}}-q^{7/2}+{\frac {19}{q^{7/2}}}+4q^{5/2}-{\frac {23}{q^{5/2}}}-9q^{3/2}+{\frac {22}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {8}{q^{11/2}}}+15{\sqrt {q}}-{\frac {20}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+3a^{5}z^{3}+4a^{5}z+a^{5}z^{-1}-3a^{3}z^{5}-8a^{3}z^{3}-8a^{3}z-a^{3}z^{-1}+az^{7}+4az^{5}-z^{5}a^{-1}+8az^{3}-2z^{3}a^{-1}+5az-2za^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-4a^{5}z^{9}-9a^{3}z^{9}-5az^{9}-6a^{6}z^{8}-17a^{4}z^{8}-20a^{2}z^{8}-9z^{8}-4a^{7}z^{7}-5a^{5}z^{7}-3a^{3}z^{7}-10az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+12a^{6}z^{6}+41a^{4}z^{6}+42a^{2}z^{6}-4z^{6}a^{-2}+10z^{6}+10a^{7}z^{5}+30a^{5}z^{5}+42a^{3}z^{5}+35az^{5}+12z^{5}a^{-1}-z^{5}a^{-3}+2a^{8}z^{4}-6a^{6}z^{4}-27a^{4}z^{4}-26a^{2}z^{4}+5z^{4}a^{-2}-2z^{4}-8a^{7}z^{3}-29a^{5}z^{3}-41a^{3}z^{3}-29az^{3}-8z^{3}a^{-1}+z^{3}a^{-3}-a^{8}z^{2}+4a^{4}z^{2}+5a^{2}z^{2}-2z^{2}a^{-2}+2a^{7}z+10a^{5}z+13a^{3}z+8az+3za^{-1}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

6

1

5

2

9

3

-6

0

11

6

5

-2

12

10

-2

-4

11

10

1

-6

8

12

4

-8

6

11

-5

-10

3

9

6

-12

1

5

-4

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$